Quantization of Drinfeld Zastava in type A

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by  Michael Finkelberg

QUANTIZATION OF DRINFELD ZASTAVA
MICHAELFINKELBERGANDLEONIDRYBNIKOV
0
1
0 Abstract. Drinfeld Zastava is a certain closure of the moduli space of maps from the
2 projective line to the Kashiwara flag scheme of the affine Lie algebra sˆln. We introduce
p an affine, reduced, irreducible, normal quiver variety Z which maps to the Zastava space
e bijectivelyatthelevelofcomplexpoints. ThenaturalPoissonstructureontheZastavaspace
S canbedescribedonZ intermsofHamiltonianreductionof acertain Poissonsubvariety of
thedualspaceofa(nonsemisimple)Liealgebra. ThequantumHamiltonianreductionofthe
3 correspondingquotientofitsuniversalenvelopingalgebraproducesaquantization Y ofthe
coordinate ringof Z. The same quantization was obtained inthe finite (as opposed to the
] affine)casegenericallyin[13]. Weprovethat,forgenericvaluesofquantizationparameters,
G
Y isaquotientoftheaffineBorelYangian.
A
.
h
t
a
m
[ 1. Introduction

1 1.1. ThemodulispaceP◦ofdegreed=(d ,d ,...,d )∈Nnbasedmapsfromtheprojective
d 0 1 n−1
v
line to the Kashiwara flag scheme of the affine Lie algebra sl admits two natural closures: an
6 n
7 affine singular Drinfeld Zastava space Zd, and a quasiprojective smooth affine Laumon space
6 Pd (see [10]). The advantageofPd lies inits smoothness(inbfact, the naturalpropermorphism
0 ̟ : P → Zd is a semismall resolution of singularities), while the advantage of Zd lies in the
d
.
9 fact that it makes sense for other simple and affine groups.
0 TheaffineLaumonspaceP isthe modulispaceoftorsionfreeparabolicsheavesonP1×P1,
d
0 and thus carries a natural Poisson structure. This structure descends to the Drinfeld Zastava
1
space Zd. We have a naturalproblemto quantize this Poissonstructure. The main goalofour
:
v note is a solution of this problem. It was already solved generically (on an open subvariety of
Xi P◦d) in the finite, i.e. d0 =0 (as opposed to the affine) case in [13].
TothisendweuseaquiverconstructionofP . Thisconstructionfollowsfromanobservation
r d
a by A. Okounkovthat P is a fixed point set component of the cyclic groupZ/nZ acting onthe
d
moduli spaceM oftorsionfree sheavesonP1×P1 framedatinfinity. The quiverinquestion
n,d
(a chainsaw quiver) is similar to but different from the A˜ quivers in Nakajima theory. In
n−1
particular, the corresponding quiver variety is not obtained by the Hamiltonian reduction of a
symplectic vectorspace. It is obtainedby the Hamiltonianreductionofa Poissonsubvarietyof
the dualvectorspaceofa(nonsemisimple)Liealgebraa withitsLie-Kirillov-Kostantbracket.
d
The correspondingcategorical(asopposedto GIT)quotientZ isreduced,irreducible,normal,
d
and admits a morphismto the Zastavaspace Zd which is bijective atthe level ofC-points. We
conjecture that this morphism is an isomorphism.
A historical comment is in order. The quiver approach to Laumon moduli spaces goes back
to S. A. Strømme [19]; we have learnt of it from A. Marian. For a more recent construction
of the monopole moduli space P◦ in the finite (as opposed to the affine) case via Hamiltonian
d
reduction see [7]. In fact, the authors of loc. cit. restrict themselves to a single open coadjoint
orbit in the Poisson subvariety of the previous paragraph.
1
2 MICHAELFINKELBERGANDLEONIDRYBNIKOV
1.2. Now the ring offunctions C[Z ]admits a naturalquantizationY asthe quantumHamil-
d d
tonian reduction of a quotient algebra of the universal enveloping algebra U(a ). The algebra
d
Y admits a homomorphism from the Borel subalgebra Y of the Yangian of type A in the
d n−1
case offinite Zastavaspace. We provethat this homomorphismis surjective. Inthe affine situ-
ation, there is a 1-parametric deformation of Z analogousto the Calogero–Moserdeformation
d
of the Hilbert scheme. This leads to the the 1-parametric family of quantum Zastava spaces,
Yµ. There is also an affine analog of Y depending on the complex parameter β (we denote
d
n
it Y ) in the same way as in [14]. There is a homomorphism Y → Yµ with β = µ+ d.
β β d l
l=1
We prove that this homomorphism is surjective for µ 6= 0. Moreover, we write down cePrtain
c c
elements in the kernel of this homomorphism and conjecture that they generate the kernel (as
a two-sided ideal). These elements are similar to the generators of the kernel of the surjective
Brundan-Kleshchevhomomorphismfromtheirshifted Yangianto afinite W-algebraoftype A.
Infact,itseemslikelythatY asafilteredalgebraisthelimitofasequenceoffiniteW-algebras
d
of type A equipped with the Kazhdan filtration.
Moreover, the similar quotients of the Borel Yangians for arbitrary simple and affine Lie
groups are likely to quantize the rings of functions on the corresponding Drinfeld Zastava
spaces.
1.3. OurmotivationforquantizationofDrinfeldZastavacamefromthefollowingsource. In[9]
weformulatedaconjectureaboutequivariantquantumcohomologyofthefiniteLaumonspaces
(it was proved recently by A. Negut). The corresponding quantum connection identifies with
the Casimir connection, and its monodromy gives rise to an action of the pure braid group
on the equivariant cohomology of P . According to the Bridgeland-Bezrukavnikov-Okounkov
d
philosophy, if we transfer this action to the equivariant K-theory via Chern character, then
it should come from an action of the pure braid group on the equivariant derived category of
coherent sheaves on P .
d
Inthe classicalcaseofNakajimaquivervarieties,therearechambers inthe spaceofstability
conditions for the GIT construction of quiver varieties, and the derived coherent categories
for the varieties in adjacent chambers are related by Kawamata-type derived equivalences.
These equivalences generate the action of the pure braid group on the derived category of a
single quiver variety. Unfortunately, this approach fails in our situation (see sections 5.1–5.4):
although we do have chambers in the space of stability conditions, the Laumonvarieties in the
adjacent chambers too often become singular and just isomorphic (as opposed to birational).
Another approach was discovered by Bezrukavnikov-Mirkovi´cin their works on localization
of g-modules in characteristic p. In our situation it works as follows: if we replace the field C
of complex numbers by an algebraic closure K of a finite field of characteristic p≫0, then the
quantizedalgebraY acquiresa big center, isomorphicto K[Z(1)](Frobenius twistof Z ). Thus
d d d
Y may be viewed as global sections of a sheaf of noncommutative algebras on Z(1). A slight
d d
upgrade of our quantization construction produces a sheaf A of noncommutative algebras on
χ
P(1) for every stability condition χ. In sections 5.5–5.8 we formulate “standard conjectures”
d
about the sheavesof algebrasA . We conjecture that they are all Morita equivalent, and their
χ
globalsectionsareisomorphictoY . Moreover,thefunctorofglobalsectionsfromthecategory
d
ofA -modulestothecategoryofY -modulesisaderivedequivalenceforχincertainchambers.
χ d
Thus, for χ in such a chamber (e.g. χ = 0), the composition of this derived equivalence with
the above Morita equivalences gives rise to an action of the pure braid group on Db(A -mod).
χ
Contrarytothe Bezrukavnikov-Mirkovi´csituation, inourcaseA is notasheafofAzumaya
χ
algebras (e.g. in the simplest case n=2, d=(0,1), we have P ≃A2, and Y is the universal
d d
QUANTIZATION OF DRINFELD ZASTAVA 3
envelopingalgebraoftheBorelsubalgebraofsl ). However,intheformalneighbourhoodofthe
2
central fiber of ̟(1) : P(1) → Z(1), the algebra A possesses a splitting module M. Tensoring
d d χ
with M defines a functor from the category of equivariant coherent sheaves on this formal
neighbourhood to the category of equivariant A -modules. We conjecture that tchis functor is
χ
a fullcembedding, and the braid group action of the previous paragraphpreserves the essential
image of this functor, thus giving rise to the braid group action on the equivariant derived
category of coherent sheaves on the formal neighbourhood of the central fiber.
1.4. Acknowledgments. We are grateful to R. Bezrukavnikov, A. Braverman, B. Feigin,
V. Ginzburg, A. Molev and V. Vologodsky for useful discussions. During the key stage of
the preparationofthispaperwehavebenefitedfromthe hospitalityandsupportoftheUniver-
sityofSydney. ThanksareduetoA.Tsymbaliukforthecarefulreadingofthefirstdraftofthis
note andspotting severalmistakes. Bothauthorswere partiallysupported by the RFBR grant
09-01-00242 and the Science Foundation of the SU-HSE awards No.T3-62.0 and 10-09-0015.
L. R. was also partially supported by the RFBR-CNRS grant 10-01-93111 and the Russian
President’s grant MK-281.2009.1.
2. A quiver approach to Drinfeld and Laumon spaces
2.1. Parabolicsheaves. WerecallthesetupofSection3of[10]. LetCbeasmoothprojective
curve of genus zero. We fix a coordinate z on C, and consider the action of C∗ on C such that
a(t)=a−1·t. We have CC∗ ={0C,∞C}. Let X be another smooth projective curve of genus
zero. We fix a coordinate y on X, and consider the action of C∗ on X such that c(x)=c−1·x.
We have XC∗ = {0X,∞X}. Let S denote the product surface C×X. Let D∞ denote the
divisor C×∞X∪∞C×X. Let D0 denote the divisor C×0X.
Given an n-tuple of nonnegative integers d = (d ,...,d ), we say that a parabolic sheaf
0 n−1
F of degree d is an infinite flag of torsion free coherent sheaves of rank n on S: ...⊂F ⊂
• −1
F ⊂F ⊂... such that:
0 1
(a) F =F (D ) for any k;
k+n k 0
(b) ch (F ) = k[D ] for any k: the first Chern classes are proportional to the fundamental
1 k 0
class of D ;
0
(c) ch (F )=d for i≡k (mod n);
2 k i
(d) F0 is locally free at D∞ and trivialized at D∞ : F0|D∞ =W ⊗OD∞;
(e) For −n ≤ k ≤ 0 the sheaf F is locally free at D , and the quotient sheaves
k ∞
Fk/F−n, F0/Fk (both supported at D0 = C ×0X ⊂ S) are both locally free at the point
∞C×0X; moreover, the local sections of Fk|∞C×X are those sections of F0|∞C×X =W ⊗OX
which take value in hw1,...,wn+ki⊂W at 0X ∈X.
The fine moduli space P of degree d parabolic sheaves exists and is a smooth connected
d
quasiprojective variety of dimension 2d +...+2d .
0 n−1
2.2. Parabolic sheaves as orbifoldsheaves. Wewillnowintroduceadifferentrealizationof
parabolicsheaves. We firstlearnedofthis constructionfromA.Okounkov,thoughitis already
presentin the workof I. Biswas[1], and goesback to M. Narasimhan. Let σ :C×X→C×X
denote the mapσ(z,y)=(z,yn), andlet Γ=Z/nZ. Then Γ acts on C×X by multiplying the
coordinate on X with the n−th roots of unity. More precisely, we choose a generator γ of Γ
whichmultipliesybyexp(2πi). WeintroduceadecreasingfiltrationW =W1 =hw ,...,w i⊃
n 1 n
W2 =hw ,...,w i⊃...⊃Wn =hw i.
2 n n
A parabolic sheaf F is completely determined by the flag of sheaves
•
F (−D )⊂F ⊂...⊂F ,
0 0 −n+1 0
4 MICHAELFINKELBERGANDLEONIDRYBNIKOV
satisfying conditions 2.1.(a–e). For −n < k ≤ 0 we consider a subsheaf F˜ ⊂ σ∗F defined as
k k
follows. Away from the line C×∞X the sheaf F˜k coincides with σ∗Fk; and the local sections
of F˜k|C×∞X are those sections of σ∗Fk|C×∞X =W ⊗OC×∞X which take value in Wk+n.
To F we can associate a single Γ-equivariant torsion free sheaf F˜ on C×X:
•
F˜ :=F˜−n+1+F˜−n+2(C×∞X−C×0X)+...+F˜0((n−1)(C×∞X−C×0X)).
Note thatF˜|C×∞X ≡W ⊗OC×∞X, andF˜|∞C×X is a trivialvector bundle, hence its trivializa-
tion on C×∞X canonically extends to a trivialization on D∞.
The sheaf F˜ will have to satisfy certain numeric and framing conditions that mimick condi-
tions 2.1.(b–e). Conversely,any Γ-equivariantsheaf F˜ that satisfies those numeric and framing
conditions will determine a unique parabolic sheaf. More precisely, for d = d +...+d ,
0 n−1
let M be the Giesecker moduli space of torsion free sheaves on C×X of rank n and second
n,d
Chern class d, trivialized on D (see [16], section 2). Then we have F˜ ∈ M . We consider
∞ n,d
the following action of Γ on W : γ(w ) = exp(2πil)w , l = 1,...,n. The action of Γ on
l n l
C×X together with its action on the trivialization at D (via the action on W) gives rise
∞
to the action of Γ on M . We have F˜ ∈ MΓ . Thus we have constructed an embedding
n,d n,d
P ֒→MΓ , F 7→F˜. The fixed point setMΓ has many connected components numbered by
d n,d • n,d
decompositions d = d +d +...+d , and the embedding P ֒→ MΓ is an isomorphism
0 1 n−1 d n,d
onto the connected component MΓ .
n,d
The inverse isomorphism takes a Γ-equivariant torsion free sheaf F˜ to the flag F (−D ) ⊂
0 0
Γ
F−n+1 ⊂...⊂F0 where for −n<k≤0 we set Fk :=σ∗ F˜ ⊗OS(kD0) .
(cid:16) (cid:17)
2.3. A quiver description of Laumon space. According to section 2 of [16], M admits
n,d
the following GIT description. We set V = Cd, and we consider M = End(V)⊕End(V)⊕
Hom(W,V)⊕Hom(V,W). A typical quadruple in M will be denoted by (A,B,p,q). We set
L ⊃ µ−1(0) := {(A,B,p,q) : AB −BA+pq = 0}. We define µ−1(0)s as the open subset of
stable quadruples, i.e. those which do not admit proper subspaces V′ ⊂ V stable under A,B
and containing p(W). The group GL(V) acts naturally on M preserving µ−1(0); its action on
µ−1(0)s is free, and M is the GIT quotient µ−1(0)s/GL(V).
n,d
In terms of this quiver realization, the action of Γ is described as follows: γ(A,B,p,q) =
(A,exp(2πi)B,exp(2πi)p,q). Recall that the action of Γ on W was desribed in 2.2: for l =
n n
1,...,n, W =hw iistheisotypiccomponentcorrespondingtothecharacterχ (γ)=exp(2πil).
l l l n
Hence the connected component of the fixed point set P ≃ MΓ admits the following quiver
d n,d
description.
We chooseanactionofΓonV suchthatthe χ -isotypiccomponentV hasdimensiond (l ∈
l l l
Z/nZ). Then MdΓ ={(Al,Bl,pl,ql)l∈Z/nZ}=
End(V )⊕ Hom(V ,V )⊕ Hom(W ,V )⊕ Hom(V ,W ):
l l l+1 l−1 l l l
l∈MZ/nZ l∈MZ/nZ l∈MZ/nZ l∈MZ/nZ
A−2 A−1 A0 A1 A2
... B−3 // V(cid:23)(cid:23) B−2 // V(cid:23)(cid:23) B−1 //(cid:24)(cid:24)V B0 //(cid:24)(cid:24)V B1 //(cid:24)(cid:24)V B2 //...
...zzpz−z2zzzzqz−W==2−(cid:15)(cid:15) 2xpx−x1xxxxqx−W;;1−(cid:15)(cid:15) 1zzpz0zzzzzq<<0W(cid:15)(cid:15)0 ||p|1|||||q==1W(cid:15)(cid:15)1 ||p|2|||||q==W2 (cid:15)(cid:15)2 |p||3||||||.>>..
−2 −1 0 1 2
(the chainsaw quiver).
QUANTIZATION OF DRINFELD ZASTAVA 5
Furthermore, µ−1(0)Γd ={(Al,Bl,pl,ql)l∈Z/nZ : Al+1Bl−BlAl+pl+1ql =0 ∀l}. Moreover,
µ−1(0)sd,Γ ={(Al,Bl,pl,ql)l∈Z/nZ ∈µ−1(0)Γd :thereisnoproperZ/nZ-gradedsubspaceV•′ ⊂V•
stable under A ,B and containing p(W )}.
• • •
Finally, the group GL(V ) acts naturally on MΓ preserving µ−1(0)Γ; its action on
l∈Z/nZ l d d
µ−1(0)s,Γ is free, and M =µ−1(0)s,Γ/ GL(V ).
d Qn,d d l∈Z/nZ l
Remark 2.4. If a point F• ∈ Pd Q≃ Mn,d has a representative (Al,Bl,pl,ql)l∈Z/nZ,
then F ∈ M has a representative (A′,B′,p′,q′) defined as follows. First of all,
0 n,d0
W′ = W ⊕ W ⊕ ... ⊕ W , V′ = V . Now A′ = A , B′ = B B ...B B , p′ =
0 1 n−1 0 0 n−1 n−2 1 0
⊕ B B ...B p , q′ =⊕ q B ...B B .
0≤l≤n−1 n−1 n−2 l l 0≤l≤n−1 l l−1 1 0
Remark 2.5. A.Neguthasintroducedin[18]themodulispacesM′ closelyrelatedtoLaumon
d
moduli spaces. Namely, M′ is defined as the moduli space of flags of locally free sheaves 0 ⊂
d
F1 ⊂...⊂Fn−1 ⊂Fn ⊂W ⊗OC such that rkFk =k, k =1,...,n; degFk =dk, and at ∞C
our flag consists of vector subbundles, and takes value hw i⊂hw ,w i⊂...hw ,...,w i⊂
1 1 2 1 n−1
W.
Let us consider the following handsaw quiver Q′
A1 A2 An−2 An−1 An
(cid:24)(cid:24)V B1 //(cid:24)(cid:24)V B2 // ... ... Bn−3// V(cid:23)(cid:23) Bn−2 // V(cid:23)(cid:23) Bn−1 //(cid:24)(cid:24)V
W ||p|1|||||q==W1 (cid:15)(cid:15)1 ||p|2|||||q==W2 (cid:15)(cid:15)2 |p||3||||||.>>.. ...ypyny−y2yyyqynyW−<<2n−(cid:15)(cid:15) 2pvnvv−v1vvvqvnvW−:: 1n(cid:15)(cid:15)−1yypynyyyyy<< n
0 1 2 n−2 n−1
with relations A B −B A +p q = 0, k = 1,...,n−1. Let M′ stand for the moduli
k+1 k k k k+1 k d
scheme of representations of Q′ (quiver with relations) such that dimW =... =dimW =
0 n−1
1, dimV = d , k =1,...,n. Let Ms′ stand for the open subscheme of stable representations
k k d
ofQ′ formedbyallthe quadruples(A ,B ,p ,q )suchthatthereisnopropergradedsubspace
• • • •
V′ ⊂V stable under A ,B andcontaining p (W ). Let G stand for the group n GL(V )
• • • • • • d k=1 k
acting onM′ naturally. Thenthe actionofG onMs′ is free,andthe argumentofSections 2.2
d d d Q
and 2.3 proves that the quotient Ms′/G is isomorphic to M′.
d d d
2.6. A quiver approach to Drinfeld Zastava. We define Z as the categorical quotient
d
µ−1(0)Γ// GL(V ), that is the spectrum of the ring of GL(V )-invariants in
d l∈Z/nZ l l∈Z/nZ l
C[µ−1(0)Γ].
d Q Q
Let χ = χ stand for the character (g ,...,g ) 7→ det(g )...det(g ) :
−1,...,−1 1 n 1 n
GL(V )→C∗. Let us denote GL(V ) by G for short.
l∈Z/nZ l l∈Z/nZ l d
Let C[µ−1(0)Γ]Gd,χr stand for the χr-isotypical component of C[µ−1(0)Γ] under the action
Q d Q d
of Gd. Then Mn,d = µ−1(0)sd,Γ/Gd = Proj r≥0C[µ−1(0)Γd]Gd,χr . We have a projective
morphism π : Mn,d →Zd. (cid:16)L (cid:17)
Let Zd stand for the Drinfeld Zastava space defined (under the name of Mα) in section 4
of [10] and (for an arbitrary almost simple simply connected group G in place of SL(n) here)
in [5]. Let ̟ : P →Zd be the morphism (semismall resolution of singularities) introduced in
d
section 5 of [10]. Our next goal is to prove the following
Theorem 2.7. a) Z is a reduced irreducible normal scheme.
d
b) The morphism ̟ : P →Zd factors as P →π Z →η Zd, and η induces a bijection between
d d d
the sets of C-points.
6 MICHAELFINKELBERGANDLEONIDRYBNIKOV
The proof occupies the rest of this section.
2.8. Examples. We consider three basic examples of Zastava spaces for the groups
SL(2),SL(3),SL(2).
2.8.1. SL(2). We take n ≥ 2, d = d = ... = d = d = 0, d = d. We have V =
c 2 3 n−1 0 1 1
V = Cd, A = A ∈ End(V), B = 0, p = p ∈ V, q = q ∈ V∗, G = GL(V). Thus
1 1 1 1 d
µ−1(0)=End(V)⊕V ⊕V∗, and Z =(End(V)⊕V ⊕V∗)//GL(V). By the classicalInvariant
d
Theory, the ring of GL(V)-invariant functions on End(V)⊕V ⊕V∗ is freely generated by the
functions a ,...,a ,b ,...,b where a :=Tr(Am), and b :=q◦Am◦p. Hence Z ≃A2d.
1 d 0 d−1 m m d
2.8.2. SL(3). We take n ≥ 3, d = d = ... = d = d = 0, d = d = 1. We have
3 4 n−1 0 1 2
V = C = V , and hence all our linear operators act between one-dimensional vector spaces,
1 2
and can be written just as numbers. We have nonzero numbers A ,A ,B ,p ,p ,q ,q , and
1 2 1 1 2 1 2
µ−1(0) is given by the single equation B (A −A )+p q = 0. The group G is just C∗ ×
1 2 1 2 1 d
C∗ with coordinates c ,c . It acts on µ−1(0) as follows: (c ,c )·(A ,A ,B ,p ,p ,q ,q ) =
1 2 1 2 1 2 1 1 2 1 2
(A ,A ,c c−1B ,c−1p ,c−1p ,c q ,c q ). The ring of C∗ ×C∗-invariant functions on µ−1(0)
1 2 1 2 1 1 1 2 2 1 1 2 2
is generated by the functions b := q p , b := q p , r := q B p , A , A with a single
1,0 1 1 2,0 2 2 2 1 1 1 2
relation b b +r(A −A )=0. Thus, Z is the product of the conifold with the affine line.
1,0 2,0 2 1 d
2.8.3. SL(2). We take n = 2, d = d = 1. We have V = C = V , and hence all our
0 1 1 2
linear operators act between one-dimensional vector spaces, and can be written just as num-
bers. cWe have nonzero numbers A ,A ,B ,B ,p ,p ,q ,q , and µ−1(0) is cut out by two
1 0 1 0 1 0 1 0
equations B (A −A )+p q = 0 = B (A − A )+p q . The group G is just C∗ ×C∗
1 0 1 0 1 0 1 0 1 0 d
with coordinates c ,c . It acts on µ−1(0) as follows: (c ,c )·(A ,A ,B ,B ,p ,p ,q ,q ) =
1 0 1 0 1 0 1 0 1 0 1 0
(A ,A ,c c−1B ,c c−1B ,c−1p ,c−1p ,c q ,c q ). The ring of C∗×C∗-invariantfunctions on
1 0 1 0 1 0 1 0 1 1 0 0 1 1 0 0
µ−1(0) is generated by the functions b := q p , b := q p , s := B B , A , A with a
1,0 1 1 0,0 0 0 0 1 1 0
single relation b b −s(A −A )2 =0.
1,0 2,0 2 1
2.9. Stratification of Z . Applying the famous Crawley-Boevey’s trick we may identify all
d
the one-dimensional spaces W, and denote the resulting line by W . Thus, W becomes the
l ∞ ∞
source of all p , and the target of all q , l∈Z/nZ:
l l
A0
(cid:24)(cid:24)
V
66 0
B−1 OO B0
q0 p0
A−1 V−1 oo p−1 // W(cid:15)(cid:15)∞ oo q1 //V(cid:25)(cid:25) 1 ww A1
22 [[ q−1 p1
B−2 ··· vv B1
The C-points of Z classify the semisimple representations of the resulting Ferris wheel quiver
d
withrelationsµ=0,tobedenotedbyQ. Moreprecisely,theC-pointsofZ classifythesemisim-
d
ple Q-modules of dimension dim=(dim(Vl)l∈Z/nZ,dim(W∞)): dim(W∞)=1, dim(Vl)=dl.
We start with the classification of simple Q-modules of dimension smaller than or equal
to dim. First suppose dim(W ) = 0. Then an irreducible module is either L (x) for some
∞ l
l ∈ Z/nZ, x ∈ C, or L(x,y) for some x ∈ C, y ∈ C∗. Here L (x) denotes the Q-module with
l
V = 0 for k 6= l, and V = C, A = x. Furthermore, L(x,y) denotes the Q-module with
k l l
V =C, A =x ∀l ∈Z/nZ, B =y.
l l l∈Z/nZ l
Q
QUANTIZATION OF DRINFELD ZASTAVA 7
Now suppose dim(W )=1. Then the irreducibility condition is equivalent to the conjunc-
∞
tion of stability condition of 2.3 and of costability: there is no proper Z/nZ-graded subspace
V′ ⊂V stableunderA ,B andcontainedinKer(q ). Wewilldenotetheopensubsetofstable
• • • • •
and costable Q-modules of dimension (1,d′) ≤ (1,d) by µ−1(0)sc,Γ. According to Chapter 2
d′
of [16], the open subsetZd′ ⊃µ−1(0)sdc′,Γ/Gd′ ⊂µ−1(0)sd,′Γ/Gd′ =Pd′ coincides with the moduli
space of locally free parabolic sheaves, to be denoted by P◦ . Thus, the isomorphism classes of
d′
irreducible Q-modules of dimension (1,d′) are parametrized by P◦ .
d′
We conclude that the set of C-points of Z is a disjoint union of the following strata. We fix
d
an n-tuple d′ ≤ d, a collection of positive integers m ,...,m , and also collections of positive
1 r
integers(ml1,...,ml,rl)l∈Z/nZ suchthatforanyl wehavedl =d′l+ ri=1mi+ rj=l 1mlj. Then
the corresponding stratum is formed by the isomorphism classes of semisimple Q-modules of
type R ⊕ ri=1L(xi,yi)⊕mi ⊕ l∈Z/nZ rj=l 1Ll(xj)⊕mlj where RP∈ P◦d′, aPnd all the pairs
(x ,y ) are distinct, and for any l all the points x , j =1,...,m , are distinct.
i i i=1,.L..,mr L L j l,rl
2.10. Dimension of µ−1(0)Γ. We consider the configuration space of Z/nZ-colored points
d
Ad :=(C−∞C)(d0)×...×(C−∞C)(dn−1). We denote µ−1(0)Γd by Md for short. We have a
morphism Υ: M →Ad sending a quadruple (A ,B ,p ,q ) to (SpecA ,...,SpecA ).
d • • • • 0 n−1
Proposition 2.11. Every fiber of Υ has dimension (d2+d ).
l∈Z/nZ l l
Proof. First we assume that dim(Υ−1(D)) = P (d2 + d ) for a colored divisor D
l∈Z/nZ l l
concentrated at one point (with colored multiplicity). We will derive the general case
P
of the proposition from this particular case by induction in d. To this end, if a divisor
D is a disjoint union of divisors D(1) and D(2) of degrees d(1) and d(2), and we know
dim(Υ−1 (D(1))) = ((d(1))2 +d(1)), dim(Υ−1 (D(2))) = ((d(2))2 +d(2)), we
d(1) l∈Z/nZ l l d(2) l∈Z/nZ l l
have to derive dim(Υ−1(D))= (d2+d ).
P l∈Z/nZ l l P
Ineffect,eachspaceV canbesplitintodirectsumV =V(1)⊕V(2),sothattheendomorphism
l P l l l
A(1) 0
A acquirestheblockdiagonalformA = l ,andSpecA(1,2) =D(1,2). Notethat
l i 0 A(2) ! l l
l
the space of such decompositions V = V(1) ⊕V(2) is an open subset in the product of two
l l l
Grassmanniansandhasdimension2d(1)d(2). Now havingwrittenthe matricesof(B ,p ,q )in
l l • • •
the block form according to our decomposition, the equation µ=0 takes the form
A(1) 0 B(11) B(12) B(11) B(12) A(1) 0 p(1)
l+1 l l − l l l + l+1 q(1) q(2)
0 A(2) ! B(21) B(22) ! B(21) B(22) ! 0 A(2) ! p(2) ! l l
l+1 l l l l l l+1 (cid:16) (cid:17)
A(1) B(11)−B(11)A(1)+p(1) q(1) A(1) B(12)−B(12)A(2)+p(1)q(2) 0 0
= l+1 l l l l+1 l l+1 l l l l+1 l = .
A(l+2)1Bl(21)−Bl(21)A(l1)+p(l+2)1ql(1) A(l+2)1Bl(22)−Bl(22)A(l2)+p(l+2)1ql(2) ! (cid:18) 0 0 (cid:19)
In particular, we see that (A(1),B(11),p(1),q(1)) (resp. (A(2),B(22),p(2),q(2))) lies in M
• • • • • • • • d(1)
(resp. in M ). So by our induction hypothesis, dim{(A(1),B(11),p(1),q(1)) : SpecA(1) =
d(2) • • • •
D(1)} = ((d(1))2 + d(1)), and dim{(A(2),B(22),p(2),q(2)) : SpecA(2) = D(2)} =
l∈Z/nZ l l • • • •
((d(2))2 +d(2)). Recall that we also have 2d(1)d(2) parameters for the choice of de-
l∈Z/nZ Pl l l l
composition V = V(1) ⊕V(2). That already gives us the desired dimension (d2+d )
P l l l l∈Z/nZ l l
altogether,anditonlyremainsto provethatthe remainingequationshaveauniquesolutionin
P
8 MICHAELFINKELBERGANDLEONIDRYBNIKOV
B(12),B(21). Itfollowsfromthefactthat,sayA(2) andA(1) havingdisjointspectra,donotad-
l l l+1 l
mitanynontrivialintertwiners,andhencethelinearmapHom(V(1),V(2))→Hom(V(1),V(2)):
l l+1 l l+1
B(21) 7→A(2) B(21)−B(21)A(1) is an isomorphism.
l l+1 l l l
Since the statement of the proposition is obvious in case d = 1, we have already
l∈Z/nZ l
proved the proposition in case D has no multiplicities (off-diagonal case). Moreover, we have
P
proved that Υ−1(Ad−∆) is smooth.
ItremainstoprovethepropositionintheoppositeextremalcasewhenDissupportedatone
point. It does not matter, which point is it, so we may and will assume it is 0. In other words,
we assume that all the endomorphisms A are nilpotent. We follow the method of G. Wilson
l
in his proof of Lemma 1.11 of [21]. Suppose first that both A and A are regular nilpotent.
l l+1
We choosebasesin V ,V sothat the matricesofA ,A areJordanblocks,andthen we see
l l+1 l l+1
that the matrix of A B −B A has the following property: for each i =1,...,min(d ,d )
l+1 l l l l l+1
the sum of all elements in the i-th diagonal (counting from the leftmost lowest corner) is 0.
Now since A B −B A = −p q has rank 1, all these min(d ,d ) diagonals must vanish
l+1 l l l l+1 l l l+1
identically. Itimposesthe followingrestrictiononthe vectorp andcovectorq writtendown
l+1 l
in our bases: the sum of numbers of the last nonzero coordinate of p and the first nonzero
l+1
coordinate of q is greater than min(d ,d ). This means that the dimension of the space of
l l l+1
all possible collections (p ,q ) is at most max(d ,d ).
l+1 l l l+1
Recall that the dimension of the space of regular nilpotent matrices A (resp. A ) is
l l+1
d2 −d (resp. d2 −d ). Furthermore, for given (A ,A ,p ,q ) the dimension of the
l l l+1 l+1 l l+1 l+1 l
space of solutions of the linear equation A B −B A = −p q equals (if it is not empty)
l+1 l l l l+1 l
the dimensionofthe spaceofintertwinersInt(A ,A ), thatismin(k,l). Altogetherweobtain
l l+1
at most d2+d2 −d −d +min(d ,d )+max(d ,d ). Summing up overall l we obtain
l l+1 l l+1 l l+1 l l+1
at most (d2+d) parameters.
l∈Z/nZ l l
NowweturntothegeneralcaseandassumethattheJordantypeofanilpotentmatrixA is
l
P
givenbyapartition(κ(l) ≥κ(l) ≥...). Let(κ(l) ≥κ(l) ≥...)standforthedualpartition. The
1 2 1 2
space of all matrices A of given type has dimension d2−(κ(l))2−(κ(l))2−.... We can choose
l l 1 2
somebasesinthe spacesV sothatthe matricesofA becomethedirectsumsofJordanblocks,
l l
andrepeattheconsiderationsoftwopreviousparagraphsblockwise. Wecometotheconclusion
thatthedimensionofthespaceofquadruples(A ,B ,p ,q )suchthattheJordantypeofA is
• • • • l
(κ(l) ≥κ(l) ≥...)is atmost (d2−(κ(l))2−(κ(l))2−...)+ i,j∈N min(κ(l),κ(l+1))+
1 2 l∈Z/nZ l 1 2 l∈Z/nZ i j
(l)
max(d ,d ). It is not hard to check (by induction in max (κ )) that this sum is
l∈Z/nZ l l+1 P P l 1
at most (d2+d ). On the other hand, the dimension of any irreducible component of
P l∈Z/nZ l l
Υ−1(d·0) cannot be less than (d2 +d) since we have already seen that the generic
P l∈Z/nZ l l
fiber of Υ has dimension (d2+d ). This completes the proof of the proposition. (cid:3)
l∈Z/nPZ l l
Corollary 2.12. M is aPn irreducible reduced complete intersection in MΓ.
d d
Proof. The complete intersection property is clear from the comparison of dimensions. It is
also clear that dimΥ−1(∆)< (d2+2d), and hence the closure of Υ−1(Ad−∆) is the
l∈Z/nZ l l
uniqueirreduciblecomponentofM . Finally,itwasshownduringtheproofofProposition2.11
d
P
thatΥ−1(Ad−∆)issmooth,andinparticular,reduced. ItfollowsfromProposition5.8.5of[12]
that M is reduced. (cid:3)
d
Remark 2.13. The subscheme Υ−1(d·0) studied in the proof of Proposition 2.11 contains
the nilcone N ⊂ M . In the situation and notations of Example 2.8.1 the nilcone N ⊂ M
d d d d
is cut out by the equations a = ... = a = 0 = b = ... = b . Equivalently, we require
1 d 0 d−1
both endomorphisms A and A+q◦p to be nilpotent. Hence N coincides with the mirabolic
d
QUANTIZATION OF DRINFELD ZASTAVA 9
nilpotent cone introduced by R. Travkin in sections 1.3 and 3.2 of [20] (under the name of Z).
The beautiful geometry of N studied in loc. cit. suggests that N might be an interesting
d d
object in itself.
2.14. Proof of Theorem 2.7.a). The categoricalquotient Z inherits the properties ofbeing
d
reduced andirreducible fromM . To prove the normalityof Z we willuse Corollary7.2 of [8].
d d
To this endwe willexhibita normalopensubschemeU ⊂Z suchthatits complementY ⊂Z
d d
is of codimension 2, and Ψ−1(Y) is of codimension 2 in M . Here Ψ: M →Z is the natural
d d d
projection. Note that M is Cohen-Macaulay (being a complete intersection), in particular, it
d
has property (S ). So all the conditions of loc. cit. will be verified, and it will guarantee the
2
normality of Z .
d
ToconstructU ⊂Z notethatthe morphismΥ: M →Ad evidentlyfactorsasM →Ψ Z →Φ
d d d d
Ad for auniquely defined morphismΦ. We introduce anopen subsetUˆ ⊂Ad formedbyallthe
colored configurations where at most 2 points collide. We set U :=Φ−1(Uˆ).
Evidently,the complement Ad−Uˆ is ofcodimension 2 in Ad, and so the codimensioncondi-
tions on U are satisfied. It remains to prove that U is normal. The argument of the first part
of the proof of Proposition 2.11 shows that after an ´etale base change in a formal neighbour-
hood of a point in Uˆ (an orderingof distinct points in a configurationin Uˆ), both Υ−1(Uˆ) and
Φ−1(Uˆ)=U decompose into a direct product of a smooth scheme, and a scheme of one of Ex-
amples 2.8.1,2.8.2, 2.8.3. Namely,Example 2.8.1occurs if twopoints ofthe same colorcollide;
Example 2.8.2 occurs if two points of different colors collide, and n>2; finally, Example 2.8.3
occurs if two points of different colors collide, and n = 2. Obviously, all the schemes of the
aboveExamplesarenormal. As normalityis stable underthe´etalebase changeandthe formal
completion, the proof of Theorem 2.7.a) is complete. (cid:3)
2.15. Proof of Theorem 2.7.b). To prove b), we recall the stratification of Zd introduced
in section 6.6 of [10]. It obviously coincides with the stratification of Z introduced in 2.9. In
d
particular, we have a bijection between the sets of C-points of Zd and Z . Moreover, for a
d
C-point s in a stratum of Zd, and the same named corresponding point in the corresponding
stratum of Z , the (reduced) fibers π−1(s) ⊂ P ⊃ ̟−1(s) coincide. In effect, they are both
d d
formed by all the parabolic sheaves with given saturation and defect in terminology of loc.
cit. Now the existence of η follows from normality of Z e.g. by the argument in the proof
d
of Proposition 2.14 of [4]. Theorem 2.7 is proved. (cid:3)
Conjecture 2.16. The morphism η : Z →Zd is an isomorphism.
d
2.17. The character of C[Z ]. Corollary 2.12 gives rise to a formula for the character of
d
C[Z ]. Let T stand for the Cartan torus of GL(W) which acts on the basis vector w via the
d k
character t , k =1,...,n. Thus T:=C∗×C∗×T acts on P via the action of the first (resp.
k d
second) copy of C∗ on C (resp. on X) via the character v (resp. u), see 2.1. The relation to
the notations of [3] is as follows: t = t2, v = v2, u = u2. Now the character of C[Z ] as a
k k d
T-module is a formalpower series in t ,...,t ,u,v whichis actually a Laurentexpansionof a
1 n
rational function to be denoted by F .
d
To calculate F we note that the action of T on C[Z ] arises from the following
d d
action of T on the symmetric algebra C[MΓ]. Let us choose a base v ,...,v
d l,1 l,dl
in V , and denote the corresponding matrix elements of A (resp. B ,p ,q ) by
l l l l l
(A(ij))1≤j≤dl (resp. (B(ij))1≤j≤dl+1, (p(i)) , (q(i)) ). Moreover, let us de-
notle b1y≤i≤Tdlthe Cartan tlorus1≤oif≤dGl actilng1o≤ni≤dal basel v1e≤cti≤ordl v via the character t .
d l,i l,i
10 MICHAELFINKELBERGANDLEONIDRYBNIKOV
Then the eigenvalues of the T × T-action on the generators of C[MΓ] are as follows:
d
Al(ij) : vtl,it−l,j1, Bl(ij) : uδ0,ltl,it−l+11,j, pl(i) : uδ1,lvtl−1t−l,i1, ql(i) : t−l 1tl,i.
The character of the T×T-action on the symmetric algebra C[MΓ] equals S :=
d d
1≤i≤dl
1≤i,j≤dl 1≤j≤dl+1 1≤i≤dl 1≤i≤dl
(1−vtl,it−l,j1)−1 (1−uδ0,ltl,it−l+11,j)−1 (1−uδ1,lvtl−1t−l,i1)−1 (1−t−l 1tl,i)−1.
l∈YZ/nZ l∈YZ/nZ l∈YZ/nZ l∈YZ/nZ
The space of equations cutting out M ⊂ MΓ has a natural base consisting of the matrix
d d
elements (E(ij))1≤j≤dl+1 of the matrices A B −B A +p q. The eigenvalue of the T×T-
l 1≤i≤dl l+1 l l l l+1 l
action on El(ij) is uδ0,lvtl,it−l+11,j. The (graded) character of the T×T-action on the external
1≤i≤dl
algebrageneratedby {(El(ij))11≤≤ji≤≤ddll+1} equals Λd :=1≤l∈jZ≤/dnlZ+1(1−uδ0,lvtl,it−l+11,j). According to
Corollary2.12, the character of the T×T-actionon C[MQ] equals S Λ . Finally, the character
d d d
Fd of the T-action on C[Zd]=C[Md]Gd equals (1,SdΛd)T where (·,·)T is the scalar product of
G -characters.
d
3. Hamiltonian reduction
3.1. Poisson structure on Laumon and Drinfeld spaces. Recall that P◦ ⊂P stands for
d d
theopensubsetoflocallyfreeparabolicsheaves. Accordingtosection5of[10],P◦ isthemoduli
d
space of based maps of degree d from (C,∞C) to the Kashiwara flag scheme of the affine Lie
algebra sl(n). According to section 1 of [5], such a moduli space of based maps is defined for
any Kac-Moody Lie algebra g; let us denote it by P◦ . In case g is a simple Lie algebra, a
g,d
symplectbic structure on P◦ was constructed in [11]. This construction applies verbatim to
g,d
P◦ for any Kac-Moody Lie algebra g, in particular for g = sl(n), and provides P◦ = P◦
g,d d sbl(n),d
with a symplectic structure Ω, and corresponding Poisson bracket {·,·} . F. Bottacin [2] has
K
generalizedthis Poissonbracketto the moduli spacesofstablebparaboliclocallyfree sheaveson
arbitrary smooth projective surfaces.
Lemma 3.2. The Poisson structure{·,·} on P◦ extends uniquely tothe same named Poisson
K d
structure on P .
d
Proof. The complement P −P◦ is a union of Cartier divisors (see e.g. section 11 of [5]). In
d d
the ´etale (x,y)-coordinates of section 3.3 of [11], these divisors are just the zero divisors of
y-coordinates. Now the explicit formula of Proposition 2 of loc. cit. shows that our bracket
{·,·} extends regularly through the generic points of these divisors. Since P is smooth, and
K d
the bivector field {·,·} is regular off codimension 2, it is regular everywhere. (cid:3)
K
Corollary 3.3. The Poisson structure {·,·} on P◦ ⊂Z extends uniquely to the same named
K d d
Poisson structure on Z .
d
Proof. The (reduced) fibers of the resolution π : P → Z were already identified with the
d d
(reduced) fibers of the resolution ̟ : P → Zd in 2.15. The latter fibers are described in
d
section 6 of [10], in particular they are connected. Due to normality of Z , the algebra of
d
functions C[Z ] coincides with the algebra C[P ]. So the Poisson bracket on C[Z ] is obtained
d d d
just as global sections of the Poisson bracket on P . (cid:3)
d